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Definition df-trkg 26166
Description: Define the class of Tarski geometries. A Tarski geometry is a set of points, equipped with a betweenness relation (denoting that a point lies on a line segment between two other points) and a congruence relation (denoting equality of line segment lengths). Here, we are using the following:
  • for congruence, (𝑥 𝑦) = (𝑧 𝑤) where = (dist‘𝑊)
  • for betweenness, 𝑦 ∈ (𝑥𝐼𝑧), where 𝐼 = (Itv‘𝑊)
With this definition, the axiom A2 is actually equivalent to the transitivity of addition, eqtrd 2853.

Tarski originally had more axioms, but later reduced his list to 11:

  • A1 A kind of reflexivity for the congruence relation (TarskiGC)
  • A2 Transitivity for the congruence relation (TarskiGC)
  • A3 Identity for the congruence relation (TarskiGC)
  • A4 Axiom of segment construction (TarskiGCB)
  • A5 5-segment axiom (TarskiGCB)
  • A6 Identity for the betweenness relation (TarskiGB)
  • A7 Axiom of Pasch (TarskiGB)
  • A8 Lower dimension axiom (DimTarskiG≥‘2)
  • A9 Upper dimension axiom (V ∖ (DimTarskiG≥‘3))
  • A10 Euclid's axiom (TarskiGE)
  • A11 Axiom of continuity (TarskiGB)
Our definition is split into 5 parts:
  • congruence axioms TarskiGC (which metric spaces fulfill)
  • betweenness axioms TarskiGB
  • congruence and betweenness axioms TarskiGCB
  • upper and lower dimension axioms DimTarskiG
  • axiom of Euclid / parallel postulate TarskiGE

So our definition of a Tarskian Geometry includes the 3 axioms for the quaternary congruence relation (A1, A2, A3), the 3 axioms for the ternary betweenness relation (A6, A7, A11), and the 2 axioms of compatibility of the congruence and the betweenness relations (A4,A5).

It does not include Euclid's axiom A10, nor the 2-dimensional axioms A8 (Lower dimension axiom) and A9 (Upper dimension axiom) so the number of dimensions of the geometry it formalizes is not constrained.

Considering A2 as one of the 3 axioms for the quaternary congruence relation is somewhat conventional, because the transitivity of the congruence relation is automatically given by our choice to take the distance as this congruence relation in our definition of Tarski geometries. (Contributed by Thierry Arnoux, 24-Aug-2017.) (Revised by Thierry Arnoux, 27-Apr-2019.)

Assertion
Ref Expression
df-trkg TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Distinct variable group:   𝑓,𝑝,𝑖,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-trkg
StepHypRef Expression
1 cstrkg 26143 . 2 class TarskiG
2 cstrkgc 26144 . . . 4 class TarskiGC
3 cstrkgb 26145 . . . 4 class TarskiGB
42, 3cin 3932 . . 3 class (TarskiGC ∩ TarskiGB)
5 cstrkgcb 26146 . . . 4 class TarskiGCB
6 vf . . . . . . . . . 10 setvar 𝑓
76cv 1527 . . . . . . . . 9 class 𝑓
8 clng 26150 . . . . . . . . 9 class LineG
97, 8cfv 6348 . . . . . . . 8 class (LineG‘𝑓)
10 vx . . . . . . . . 9 setvar 𝑥
11 vy . . . . . . . . 9 setvar 𝑦
12 vp . . . . . . . . . 10 setvar 𝑝
1312cv 1527 . . . . . . . . 9 class 𝑝
1410cv 1527 . . . . . . . . . . 11 class 𝑥
1514csn 4557 . . . . . . . . . 10 class {𝑥}
1613, 15cdif 3930 . . . . . . . . 9 class (𝑝 ∖ {𝑥})
17 vz . . . . . . . . . . . . 13 setvar 𝑧
1817cv 1527 . . . . . . . . . . . 12 class 𝑧
1911cv 1527 . . . . . . . . . . . . 13 class 𝑦
20 vi . . . . . . . . . . . . . 14 setvar 𝑖
2120cv 1527 . . . . . . . . . . . . 13 class 𝑖
2214, 19, 21co 7145 . . . . . . . . . . . 12 class (𝑥𝑖𝑦)
2318, 22wcel 2105 . . . . . . . . . . 11 wff 𝑧 ∈ (𝑥𝑖𝑦)
2418, 19, 21co 7145 . . . . . . . . . . . 12 class (𝑧𝑖𝑦)
2514, 24wcel 2105 . . . . . . . . . . 11 wff 𝑥 ∈ (𝑧𝑖𝑦)
2614, 18, 21co 7145 . . . . . . . . . . . 12 class (𝑥𝑖𝑧)
2719, 26wcel 2105 . . . . . . . . . . 11 wff 𝑦 ∈ (𝑥𝑖𝑧)
2823, 25, 27w3o 1078 . . . . . . . . . 10 wff (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))
2928, 17, 13crab 3139 . . . . . . . . 9 class {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}
3010, 11, 13, 16, 29cmpo 7147 . . . . . . . 8 class (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
319, 30wceq 1528 . . . . . . 7 wff (LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
32 citv 26149 . . . . . . . 8 class Itv
337, 32cfv 6348 . . . . . . 7 class (Itv‘𝑓)
3431, 20, 33wsbc 3769 . . . . . 6 wff [(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
35 cbs 16471 . . . . . . 7 class Base
367, 35cfv 6348 . . . . . 6 class (Base‘𝑓)
3734, 12, 36wsbc 3769 . . . . 5 wff [(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
3837, 6cab 2796 . . . 4 class {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}
395, 38cin 3932 . . 3 class (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})})
404, 39cin 3932 . 2 class ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
411, 40wceq 1528 1 wff TarskiG = ((TarskiGC ∩ TarskiGB) ∩ (TarskiGCB ∩ {𝑓[(Base‘𝑓) / 𝑝][(Itv‘𝑓) / 𝑖](LineG‘𝑓) = (𝑥𝑝, 𝑦 ∈ (𝑝 ∖ {𝑥}) ↦ {𝑧𝑝 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})}))
Colors of variables: wff setvar class
This definition is referenced by:  axtgcgrrflx  26175  axtgcgrid  26176  axtgsegcon  26177  axtg5seg  26178  axtgbtwnid  26179  axtgpasch  26180  axtgcont1  26181  tglng  26259  f1otrg  26584  eengtrkg  26699
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